2.10 The Starry Messenger

“Let God look and judge!”

Cardinal
Humbert, 1054 AD

Maxwell's equations are
very successful at describing the propagation of light based on the model of
electromagnetic waves, not only in material media but also in a vacuum, which
is considered to be a region free of material substances. According to this
model, light propagates in vacuum at a speed ,
where m0 is the
permeability constant and e0 is the permittivity of the vacuum, defined in terms
of Coulombs law for electrostatic force

The SI system of units is
defined so that the permeability constant takes on the value m0 = 4p10-7 tesla meter per ampere, and we can measure the value
of the permittivity (typically by measuring the capacitance C between
parallel plates of area A separated by a distance d, using the relation e0 =
Cd/A) to have the value e0 = (8.854187818)10-12 coulombs2
per newton meters2. This leads to the familiar value

for the speed of light in a
vacuum. Of course, if we place some substance between our capacitors when
determining e0 we
will generally get a different value, so the speed of light is different in
various media. This leads to the index of refraction of various transparent
media, defined as n = cvacuum / cmedium. Thus Maxwell's
theory of electro-magnetism seems to clearly imply that the speed of
propagation of such electromagnetic waves depends only on the medium, and is
independent of the speed of the source.

On the other hand, it also
suggests that the speed of light depends on the motion of the medium, which
is easy to imagine in the case of a material medium like glass, but not so
easy if the "medium" is the vacuum of empty space. How can we even
assign a state of motion to the vacuum? In struggling to answer this
question, people tried to imagine that even the vacuum is permeated with some
material-like substance, the ether, to which a definite state of motion could
be assigned. On this basis it was natural to suppose that Maxwell's equations
were strictly applicable (and the speed of light was exactly c) only
with respect to the absolute rest frame of the ether. With respect to other
frames of reference they expected to find that the speed of light differed,
depending on the direction of travel. Likewise we would expect to find
corresponding differences and anisotropies in the capacitance of the vacuum
when measured with plates moving at high speed relative to the ether.

However, when extremely
precise interferometer measurements were carried out to find a directional
variation in the speed of light on the Earth's surface (presumably moving
through the ether at fairly high speed due to the Earth's rotation and its
orbital motion around the Sun), essentially no directional variation in light
speed was found that could be attributed to the motion of the apparatus
through the ether. Of course, it had occurred to people that the ether might
be "dragged along" by the Earth, so that objects on the Earth's
surface are essentially at rest in the local ether. However, these
"convection" hypotheses are inconsistent with other observed
phenomena, notably the aberration of starlight, which can only be explained
in an ether theory if it is assumed that an observer on the Earth's surface
is not at rest with respect to the local ether. Also, careful
terrestrial measurements of the paths of light near rapidly moving massive
objects showed no sign of any "convection". Considering all this,
the situation was considered to be quite puzzling.

There is a completely
different approach that could be taken to modeling the phenomena of light,
provided we're willing to reject Maxwell's theory of electromagnetic waves,
and adopt instead a model similar to the one that Newton often
seemed to have in mind, namely, an "emission theory". One advocate
of such a theory early in the early 1900's was Walter Ritz, who rejected
Maxwell's equations on the grounds that the advanced potentials allowed by
those equations were unrealistic. Ritz debated this point with Albert
Einstein, who argued that the observed asymmetry between advanced and
retarded waves is essentially statistical in origin, due to the improbability
of conditions needed to produce coherent advanced waves. Neither man
persuaded the other. (Ironically, Einstein himself had already posited that
Maxwell's equations were inadequate to fully represent the behavior of light,
and suggested a model that contains certain attributes of an emission theory
to account for the photo-electric effect, but this challenge to Maxwell's
equations was on a more subtle and profound level than Ritz's objection to
advanced potentials.)

In place of Maxwell's
equations and the electromagnetic wave model of light, the advocates of
"emission theories" generally assume a Galilean or Newtonian
spacetime, and postulate that light is emitted and propagates away from the
source (perhaps like Newtonian corpuscles) at a speed of c relative to the
source. Thus, according to emission theories, if the source is moving
directly toward or away from us with a speed v, then the light from that
source is approaching us with a speed c+v or c-v
respectively. Naturally this class of theories is compatible with experiments
such as the one performed by Michelson and Morley, since the source of the
light is moving along with the rest of the apparatus, so we wouldn't expect
to find any directional variation in the speed of light in such experiments. Also,
an emission theory of light is compatible with stellar aberration, at least
up to the limits of observational resolution. In fact, James Bradley (the
discoverer of aberration) originally explained it on this very basis.

Of course, even an emission
theory must account for the variations in light speed in different media,
which means it can't simply say that the speed of light depends only
on the speed of the source. It must also be dependent on the medium through
which it is traveling, and presumably it must have a "terminal
velocity" in each medium, i.e., a certain characteristic speed that it
can maintain indefinitely as it propagates through the medium. (Obviously we
never see light come to rest, nor even do we observe noticeable "slowing"
of light in a given medium, so it must always exhibit a characteristic
speed.) Furthermore, based on the principles of an emission theory, the
medium-dependent speed must be defined relative to the rest frame of the
medium.

For example, if the
characteristic speed of light in water is cw, and a body of water
is moving relative to us with a speed v, then (according to an emission
theory) the light must move with a speed cw + v relative to us
when it travels for some significant distance through that water, so that it
has reached its "steady-state" speed in the water. In optics this
distance is called the "extinction distance", and it is known to be
proportional to 1/(rl), where r is the density of
the medium and l is the wavelength of light. The extinction distance
for most common media for optical light is extremely small, so essentially
the light reaches its steady-state speed as soon as it enters the medium.

An experiment performed by
Fizeau in 1851 to test for optical "convection" also sheds light on
the viability of emission theories. Fizeau sent beams of light in both
directions through a pipe of rapidly moving water to determine if the light
was "dragged along" by the water. Since the refractive index of
water is about n = c/cw = 1.33 where cw is the speed of
light in water, we know that cw equals c/1.33, which is about 75%
of the speed of light in a vacuum. The question is, if the water is in motion
relative to us, what is the speed (relative to us) of the light in the water?

If light propagates in an
absolutely fixed background ether, and isn't dragged along by the water at
all, we would expect the light speed to still be cw relative to
the fixed ether, regardless of how the water moves. This is admittedly a
rather odd hypothesis (i.e., that light has a characteristic speed in water,
but that this speed is relative to a fixed background ether, independent of
the speed of the water), but it is one possibility that can't be ruled out a
priori. In this case the difference in travel times for the two
directions would be proportional to

which implies no phase
shift in the interferometer. On the other hand, if emission theories are
right, the speed of the light in the water (which is moving at the speed v)
should be cw+v in the direction of the water's motion, and cw-v in the opposite direction. On this basis the difference in travel
times would be proportional to

This is a very small amount
(remembering that cw is about 75% of the speed of light in a
vacuum), but it is large enough that it would be measurable with delicate
interferometry techniques.

The results of Fizeau's
experiment turned out to be consistent with neither of the above
predictions. Instead, he found that the time difference (proportional to the
phase shift) was a bit less than 43.5% of the prediction for an emission
theory (i.e., 43.5% of the prediction based on the assumption of complete
convection). By varying the density of the fluid we can vary the refractive
index and therefore cw, and we find that the measured phase shift
always indicates a time difference of (1-cw2)
times the prediction of the emission theory. For water we have cw
= 0.7518, so the time lag is (1-cw2)
= 0.4346 of the emission theory prediction.

This implies that if we let
S(cw,v) and S(cw,-v) denote the speeds
of light in the two directions, we have

By partial fraction
decomposition this can be written in the form

where

Also, in view of the
symmetry S(u,v) = S(v,u), we can swap cw with v to give

Solving these last two
equations for A and B gives A = 1 - vcw
and B = 1 + vcw, so the function S is

which of course is the
relativistic formula for the composition of velocities. So, even if we
rejected Maxwell's equations, it still appears that emission theories cannot
be reconciled with Fizeau's experimental results.

More evidence ruling out
simple emission theories comes from observations of a supernova made by
Chinese astronomers in the year 1054 AD. When a star explodes as a supernova,
the initial shock wave moves outward through the star's interior in just
seconds, and elevates the temperature of the material to such a high level
that fusion is initiated, and much of the lighter elements are fused into
heavier elements, including some even heavier than iron. (This process yields
most of the interesting elements that we find in the world around us.) Material
is flung out at high speeds in all directions, and this material emits
enormous amounts of radiation over a wide range of frequencies, including
x-rays and gamma rays. Based on the broad range of spectral shifts (resulting
from the Doppler effect), it's clear that the sources of this radiation have
a range of speeds relative to the Earth of over 10000 km/sec. This is because
we are receiving light emitted by some material that was flung out from the
supernova in the direction away from the Earth, and by other material that
was flung out in the direction toward the Earth.

If the supernova was
located a distance D from us, then the time for the "light" (i.e.,
EM radiation of all frequencies) to reach us should be roughly D/c, where c
is the speed of light. However, if we postulate that the actual speed of the
light as it travels through interstellar space is affected by the speed of
the source, and if the source was moving with a speed v relative to the Earth
at the time of emission, then we would conclude that the light traveled at a
speed of c+v on it's journey to the Earth. Therefore, if the sources of light
have velocities ranging from -v to +v, the first light from the initial explosion
to reach the Earth would arrive at the time D/(c+v), whereas the last light
from the initial explosion to reach the Earth would arrive at D/(c-v). This
is illustrated in the figure below.

Hence the arrival times for
light from the initial explosion event would be spread out over an interval
of length D/(c-v) - D/(c+v), which equals
(D/c)(2v/c) / (1-(v/c)2). The denominator is virtually 1,
so we can say the interval of arrival times for the light from the explosion
event of a supernova at a distance D is about (D/c)(2v/c), where v is the
maximum speed at which radiating material is flung out from the supernova.

However, in actual
observations of supernovae we do not see this "spreading
out" of the event. For example, the Crab supernova was about 6000 light
years away, so we had D/c = 6000 years, and with a range of source speeds of
10000 km/sec (meaning v = ±5000) we would expect a range
of arrival times of 200 years, whereas in fact the Crab was only bright for
less than a year, according to the observations recorded by Chinese
astronomers in July of 1054 AD. For a few weeks the "guest star", as
they called it, in the constellation Taurus was the brightest star in the
sky, and was even visible in the daytime for twenty-six days. Within two
years it had disappeared completely to the naked eye. (It was not visible in Europe or the
Islamic countries, since Taurus is below the horizon of the night sky in July
for northern latitudes.) In the time since the star went supernova the debris
has expanded to it's present dimensions of about 3 light years, which implies
that this material was moving at only (!) about 1/300 the speed of light. Still,
even with this value of v, the bright explosion event should have been
visible on Earth for about 40 years (if the light really moved through space
at c ± v). Hence we can conclude that the light actually
propagated through space at a speed essentially independent of the speed of
the sources.

However, although this
source independence of light speed is obviously consistent with Maxwell's
equations and special relativity, we should be careful not to read too much
into it. In particular, this isn't direct proof that the speed of light in a
vacuum is independent of the speed of the source, because for visible light
(which is all that was noted on Earth in July of 1054 AD) the extinction
distance in the gas and dust of interstellar space is much less than the 6000
light year distance of the Crab nebula. In other words, for visible light,
interstellar space is not a vacuum, at least not over distances of many light
years. Hence it's possible to argue that even if the initial speed of light
in a vacuum was c+v, it would have slowed to c for most of its journey to
Earth. Admittedly, the details of such a counter-factual argument are lacking
(because we don't really know the laws of propagation of light in a universe
where the speed of light is dependent on the speed of the source, nor how the
frequency and wavelength would be altered by interaction with a medium, so we
don't know if the extinction distance is even relevant), but it's not totally
implausible that the static interstellar dust might affect the propagation of
light in such a way as to obscure the source dependence, and the extinction
distance seems a reasonable way of quantifying this potential effect.

A better test of the
source-independence of light speed based on astronomical observations is to
use light from the high-energy end of the spectrum. As noted above, the
extinction distance is proportional to 1/(rl). For some
frequencies of x-rays and gamma rays the extinction distance in interstellar
space is about 60000 light years, much greater than the distances to many
supernova events, as well as binary stars and other configurations with
identifiable properties. By observing these events and objects it has been
found that the arrival times of light are essentially independent of
frequency, e.g., the x-rays associated with a particular identifiable event
arrive at the same time as the visible light for that event, even though the
distance to the event is much less than the extinction distance for x-rays. This
gives strong evidence that the speed of light in a vacuum is actually
invariant and independent of the motion of the source.

With the aid of modern spectroscopy
we can now examine supernovae events in detail, and it has been found that
they exhibit several characteristic emission lines, particularly the
signature of atomic hydrogen at 6563 angstroms. Using this as a marker we can
determine the Doppler shift of the radiation, from which we can infer the
speed of the source. The energy emitted by a star going supernova is
comparable to all the energy that it emitted during millions or even billions
of years of stable evolution. Three main categories of supernovae have been
identified, depending on the mass of the original star and how much of its
"nuclear fuel" remains. In all cases the maximum luminosity occurs
within just the first few days, and drops by 2 or 3 magnitudes within a month,
and by 5 or 6 magnitudes within a year. Hence we can conclude that the light
actually propagated through empty space at a speed essentially
independent of the speed of the sources.

Another interesting
observation involving the propagation of light was first proposed in 1913 by
DeSitter. He wondered whether, if we assume the speed of light in a vacuum is
always c with respect to the source, and if we assume a Galilean spacetime,
we would notice anything different in the appearances of things. He
considered the appearance of binary star systems, i.e., two stars that orbit
around each other. More than half of all the visible stars in the night sky
are actually double stars, i.e., two stars orbiting each other, and the
elements of their orbits may be inferred from spectroscopic measurements of
their radial speeds as seen from the Earth. DeSitter's basic idea was that if
two stars are orbiting each other and we are observing them from the plane of
their mutual orbit, the stars will be sometimes moving toward the Earth
rapidly, and sometimes away. According to an emission theory this orbital
component of velocity should be added to or subtracted from the speed of
light. As a result, over the hundreds or thousands of years that it takes the
light to reach the Earth, the arrival times of the light from approaching and
receding sources would be very different.

Now, before we go any
further, we should point out a potential difficulty for this kind of
observation. The problem (again) is that the "vacuum" of empty
space is not really a perfect vacuum, but contains small and sparse particles
of dust and gas. Consequently it acts as a material and, as noted above,
light will reach it's steady-state velocity with respect to that interstellar
dust after having traveled beyond the extinction distance. Since the
extinction distance for visible light in interstellar space is quite short,
the light will be moving at essentially c for almost its entire travel time,
regardless of the original speed. For this reason, it's questionable whether
visual observations of celestial objects can provide good tests of emission
theory predictions. However, once again we can make use of the high-frequency
end of the spectrum to strengthen the tests. If we focus on light in the
frequency range of, say, x-rays and gamma rays, the extinction distance is
much larger than the distances to many binary star systems, so we can carry
out DeSitter's proposed observation (in principle) if we use x-rays, and this
has actually been done by Brecher in 1977.

With the proviso that we
will be focusing on light whose extinction distance is much greater than the
distance from the binary star system to Earth (making the speed of the light
simply c plus the speed of the star at the time of emission), how should we
expect a binary star system to appear? Let's consider one of the stars in the
binary system, and write its coordinates and their derivatives as

where D is the distance
from the Earth to the center of the binary star system, R is the radius of
the star's orbit about the system's center, and w is the angular speed of the
star. We also have the components of the emissive light speed

c2 = cx2 + cy2

In these terms we can write
the components of the absolute speed of the light emitted from the star at
time t:

Now, in order to reach the
Earth at time T the light emitted at time t must travel in the x direction
from x(t) to 0 at a speed of for a
time Dt = T-t, and similarly for the y direction. Hence we have

Substituting for x, y, and
the light speed derivatives , , we have

Squaring both sides of both
equations, and adding the resulting equations together, gives

Re-arranging terms gives
the quadratic in Dt

If we define the normalized
parameters

then the quadratic in Dt becomes

Solving this quadratic for
Dt
= T-t and then adding t to both sides gives the
arrival time T on Earth as a function of the emission time t on the star

If the star's speed v is
much less than the speed of light, this can be expressed very nearly as

The derivative of T with
respect to t is

and this takes it's minimum
value when t = 0, where we have

Consequently we find the
DeSitter effect, i.e., dT/dt goes negative if d > r / v2. Now,
we know from Kepler's third law (which also applies in relativistic gravity
with the appropriate choice of coordinates) that m = r3 w2
= r v2, so we can substitute m/r for v2 in our
inequality to give the condition d > r2 / m. Thus if the
distance of the binary star system from Earth exceeds the square of the
system's orbital radius divided by the system's mass (in geometric units) we
would expect DeSitter's apparitions - assuming the speed of light is c ± v.

As an example, for a binary
star system a distance of d = 20000 light-years away, with an orbital radius
of r = 0.00001 light-years, and an orbital speed of v = 0.00005, the arrival
time of the light as a function of the emission time is as shown below:

This corresponds to a star
system with only about 1/6 solar mass, and an orbital radius of about 1.5
million kilometers. At any given reception time on Earth we can typically
"see" at least three separate emission events from the same star at
different points in its orbit. These ghostly apparitions are the effect that
DeSitter tried to find in photographs of many binary star systems, but none
exhibited this effect. He wrote

The observed velocities of
spectroscipic doubles are as a matter of fact satisfactorily represented by a
Keplerian motion. Moreover in many cases the orbit derived from the radial
velocities is confirmed by visual observations (as for d Equuli, z
Herculis, etc.) or by eclipse observations (as in Algol variables). We can
thus not avoid the conclusion [that] the velocity of light is independent of
the motion of the source. Ritz’s theory would force us to assume that the
motion of the double stars is governed not by Newton’s law, but by a much
more complicated law, depending on the star’s distance from the earth, which
is evidently absurd.

Of course, he was looking
in the frequency range of visible light, which we've noted is subject to extinction.
However, in the x-ray range we can (in principle) perform the same basic
test, and yet we still find no traces of these ghostly apparitions in binary
stars, nor do we ever see the stellar components going in "reverse
time" as we would according to the above profile. (Needless to say, for
star systems at great distances it is not possible to distinguish the changes
in transverse positions but, as noted above, by examining the Doppler shift
of the radial components of their motions we can infer the motions of the
individual bodies.) Hence these observations support the proposition that the
speed of light in empty space is essentially independent of the speed of the
source.

In comparison, if we take
the relativistic approach with constant light speed c, independent of the
speed of the source, an analysis similar to the above gives the approximate
result

whose derivative is

which is always positive
for any v less than 1. This means we can't possibly have images arriving in
reverse time, nor can we have any multiple appearances of the components of
the binary star system.

Regarding this subject,
Robert Shankland recalled Einstein telling him (in 1950) that he had himself
considered an emission theory of light, similar to Ritz's theory, during the
years before 1905, but he abandoned it because

he could think of no form of
differential equation which could have solutions representing waves whose
velocity depended on the motion of the source. In this case the emission
theory would lead to phase relations such that the propagated light would be
all badly "mixed up" and might even "back up on itself". He
asked me, "Do you understand that?" I said no, and he carefully
repeated it all. When he came to the "mixed up" part, he waved his
hands before his face and laughed, an open hearty laugh at the idea!